Centered-potential regularization for the advection upstream splitting method

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upstream splitting method

Martin Parisot, Jean-Paul Vila

To cite this version:

Martin Parisot, Jean-Paul Vila. Centered-potential regularization for the advection upstream splitting method . SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2016.

�hal-01152395v2�

Vol. n, No. p, pp. 0000–0000

CENTERED-POTENTIAL REGULARIZATION FOR ADVECTION UPSTREAM SPLITTING METHOD

^∗

MARTIN PARISOT^† AND JEAN-PAUL VILA^‡

Abstract. The current paper is devoted to a centered IMEX scheme in multi-dimensional framework for a wide class of multicomponent and isentropic flows. The proposed strategy is based on a regularized model where the advection velocity is modified by the gradient of the potential of the conservative forces in both mass and momentum equations. The stability of the scheme is ensured by the dissipation of mechanic energy, which stands for a mathematical entropy, under an advective CFL condition. The main physical properties such as positivity, conservation of the total momentum and conservation of the steady state at rest are satisfied. In addition, asymptotic preserving properties in the regimes (‘incompressible’ and ‘acoustic’) are analyzed. Finally, several simulations are presented to illustrate our results in a simplified context of oceanic flows in one dimension.

Key words. conservation laws, low-Mach number, low-Froude number, well-balanced scheme, entropy dissipation, asymptotic preserving

AMS subject classifications. 35L60, 76M12, 86A05, 76E20, 35B40 DOI.XXXXX

1. Introduction. In the current paper, we are interested in a system of con- servation laws with a large number L of unknowns ρ

_i

(t, X) ∈ R

+

/{0} satisfying an advection equation with their own velocity U

_i

(t, X) ∈ R

^d

in a multidimensional fram- work d ∈ N \ {0}, where X ∈ Ω ⊂ R

^d

is the space variable and t ≥ 0 is the time variable. The space domain is assumed to be periodic and its measure denoted by |Ω|

is finite. The force field applied to the fluids is supposed to be irrotational, defined by the scalar potential φ

i

(ρ, X) ∈ R with ρ = (ρ

1

, . . . , ρ

L

)

^|

. The advection model reads

(M

_ε^t

)







∂

t

ρ

i

+ ∇ · (ρ

i

U

i

) = 0

∂

t

(ρ

i

U

i

) + ∇ · (ρ

i

U

i

⊗ U

i

) = −ρ

i

∇ φ

i

ε

²

with a given initial condition ρ

i

(0, X) = ρ

⁰_i

(X) > 0 and U

i

(0, X ) = U

_i⁰

(X ). In the following, we refer to the scalar unknowns ρ

i

as the masses and to the second equation of (M

_ε^t

) as the momentum balance. The parameter ε is the dimensionless number characterizing the ratio of the inertial and potential forces. In addition, the potential energy E (ρ, X) is defined by ∂

_ρi

E := φ

_i

, the kinetic energy by K

i

:=

^ρ₂ⁱ

kU

i

k

²

and the mechanical energy of the model by E :=

_ε^E2

+ P

L

i=1

K

i

. In order to lead to physical solutions, for any positive mass ρ

i

> 0, the potential energy should be a strictly convex function of the masses. More precisely, the Hessian H (ρ, X) of the potential energy E, defined by H

_ij

= ∂

²_ρ

jρ_i

E = ∂

_ρj

φ

_i

has to be positive-definite and the model

∗Received by the editors XXXXX; accepted for publication (in revised form) XXXXX; published electronically XXXX.

http://www.siam.org/journals/sinum/XXXXX.html

†INRIA Paris, ANGE Project-Team, 75589 Paris Cedex 12, France; CEREMA, ANGE Project- Team, F-60280 Margny-L`es-Compi`egne, France; LJLL, UPMC Universit´e Paris VI, Sorbonne Uni- versit´es, UMR CNRS 7958, F-75005 Paris, France; supported by the SHOM grant 15CR02 (mar- [email protected])

‡Institut de Math´ematiques de Toulouse, UMR CNRS 5219, INSA, F-31077 Toulouse, France ([email protected])

0

(M

_ε^t

) should satisfy the following entropy inequality, corresponding to the second law of thermodynamics

(1.1) ∂

t

E +

L

X

i=1

∇ ·

ρ

i

φ

i

ε

²

+ K

i

U

i

≤ 0.

The solution of (M

_ε^t

) satisfies many steady states, which are generally hard to estimate. As is usual in the literature [2, 4, 17, 34, 38], only the steady state at rest is considered, i.e. with the additional condition u

i

= 0.

Proposition 1.1. The state defined by

u

_i

= 0 and φ

_i

(ρ (t, X) , X ) = φ

_i

is a steady state of the model (M

_ε^t

), the so-called steady state at rest.

Note that several works present stability results for particular steady states with non-vanishing velocity, see for example [8, 22, 35].

In the current work, a particular attention is given to the resolution of the advec- tion model (M

_ε^t

) in the regime ε 1. In such a regime, the main term of the potential is constant and is given by φ

_i

=

_|Ω|¹

R

Ω

φ

i

ρ

⁰

(X) , X

dX and the main term of the mass denoted by ρ

_i

(x) satisfies φ

_i

= φ

i

(ρ (X ) , X). Two asymptotic models seem relevant, according to the time scale.

• For a short time scale t = O

ε→0

(ε), using the rescaled time τ =

_ε^t

and assuming that the initial condition satisfies the scaling U

_i⁰

(X) = O (ε) and ρ

⁰_i

= ρ

_i

+ O (ε), the advection model (M

_ε^t

) tends when ε vanishes to a system of wave equations usually called acoustic regime defined by

(M

₀^τ

)



 

 

∂

_τ

φ e

_i

+

L

X

j=1

H

_ij

∇ · ρ

_j

U e

_j

= 0

∂

_τ

U e

_i

+ ∇ φ e

_i

= 0 with the initial condition given by φ e

_i

(0, X) = P

L

j=1

H

_ij

ρ

⁰_j

(X ) − ρ

_j

(X) and U e

_i

(0, X) =

^Ui0_ε^(X)

. The main part of the Hessian is given by H

_ij

= H

_ij

(ρ (X ) , X ).

For more details see [43].

• For a long time scale t = O

ε→0

(1), assuming that the Hessian of the potential energy is well-conditioned independently of ε, i.e. κ = O

ε→0

(1) with κ the condition number of H and the initial condition satisfies the scaling ρ

⁰_i

= ρ

_i

+ O (ε) and ∇ · ρ

⁰_i

U

_i⁰

= O (ε), the advection model (M

_ε^t

) tends to a divergence free model usually called incompressible regime when ε vanishes, i.e.

(M

₀^t

)

∇ · ρ

_i

U

_i

= 0

∂

t

U

i

+ U

i

· ∇U

i

= −∇ψ

i

with the initial condition U

_i

(0, X) = U

_i⁰

(X ). The unknown ψ

_i

(t, X) acts as a

Lagrange multiplier to satisfy the divergence free condition. See [23, 24, 28] for the

case L = 1.

Proposition 1.2. Assume that the potential energy does not depend on the space variable, i.e. E (ρ, X ) = E (ρ). Then the total momentum satisfies the following conservation law

∂

_t

L

X

i=1

ρ

_i

U

_i

! + ∇ ·

L

X

i=1

ρ

_i

U

_i

⊗ U

_i

+ P (ρ) ε

²

I

_d

!

= 0.

with P (ρ) = P

L

i=1

ρ

_i

φ

_i

(ρ) − E (ρ).

Many physical systems satisfy the assumptions of (1.1), Proposition 1.1 and the scaling of the Hessian κ = O

ε→0

(1). In the case of the one component system, i.e.

L = 1, the isentropic Euler equations [41, 44, 42, 15, 14] with or without gravity force can be written under the form (M

_ε^t

). The isentropic pressure is a function of the mass and can be linked to the potential and the potential energy thanks to the relation P = ρφ − E. In the case of polytropic fluids, where the pressure is given by P = λρ

^γ

with λ > 0 and the adiabatic coefficient γ > 1, the potential energy, the potential and the Hessian are respectively given by

E = λ ρ

^γ

γ − 1 + ρgz, φ = λ γρ

^γ−1

γ − 1 + gz and H = λγρ

^γ−2

where z is the vertical coordinate. In this context, the dimensionless parameter ε corresponds to the so-called Mach number. In the case without gravity force, i.e. g = 0, it is clear that the assumption of Proposition 1.2 is fulfilled. For a multicomponent system, i.e. L > 1, the most obvious physical system which can be written under the form (M

_ε^t

) is the mixture model [3]. Several numerical strategies can be cited to adapt the Riemann solvers to the non-conservative hyperbolic equations, see [6, 9, 32].

However, the eigenvalues, generally hard to approximate in the case of a large number of equations, are required for many numerical resolution of hyperbolic equations.

In the framework of free surface flow, the classical shallow water model [11] as well as the multilayer model [1, 27, 36] can be written under the form (M

_ε^t

). In this context, the so-called mass ρ

i

> 0 corresponds to the effective mass of the i

^th

water column %

i

h

i

with %

i

the density of the fluid and h

i

the layer thickness. The velocity U

i

corresponds to the mean horizontal velocity. In this context, the dimensionless parameter ε corresponds to the so-called Froude number. Adopting the convention that the layers are numbered from the free surface to the bottom, the potential is given by

φ

_i

= g



B +

L

X

j=1

%

j

h

j

%

_max(i,j)





with B (X) the bottom elevation. The potential energy and the Hessian are respec- tively given by

E = g

L

X

i=1







B + 1 2

L

X

j=1

%

_j

h

_j

%

_max(i,j)



 %

_i

h

_i



 , and H

_ij

= g

%

_max(i,j)

.

Note that the Hessian is positive-definite iff the layers are well-stratified, i.e. iff

%

_i

< %

_i+1

. This assumption is a necessary (not sufficient) condition to ensure the

multilayer shallow water model is hyperbolic, see [13, 30, 31]. Finally, in the case of flat topography, i.e. B = 0, it is clear that the assumption of Proposition 1.2 is satisfied and leads in this context to the conservation of the momentum of the column of water.

Classical Riemann solvers are well-known to be to much dissipative in the asymp- totic regime ε 1, and require a very restrictive CFL condition, see [19]. More precisely, the asymptotic scheme obtained by passing ε to zero in classical Riemann solvers is not consistent with the acoustic regime (M

₀^τ

), see [12]. The purpose of this work is to propose a numerical strategy (S

_ε^δt

) consistent with (M

_ε^t

) for any value of ε and satisfying the main physical properties (1.1), Proposition 1.2 and Proposition 1.1.

In particular, the asymptotic scheme (S

₀^δt

) obtained by passing ε to zero in the numer- ical scheme (S

_ε^δt

), is consistent with the incompressible regime (M

₀^t

). Similarly, for small time step δ

_t

= εδ

_τ

, the asymptotic scheme (S

₀^δτ

) obtained by passing ε to zero in the numerical scheme S

^δ_ε^τ

, is consistent with the incompressible regime (M

₀^τ

).

These properties, called in the literature asymptotic preserving (AP), see [21], can be illustrated by Fig. 1.

(S

_ε^δt

)

(M

_ε^t

)

(S

₀^δτ

) (M

₀^τ

)

(S

₀^δt

)

(M

₀^t

)

δx→0 δ_x→0

δ_t=εδ_τ ε→0

t=ετ ε→0

δt= O

ε→0(1) ε→0

t= O

ε→0(1) ε→0

δ_x→0

Fig. 1.Asymptotic preserving properties of the CPR scheme (S^δε^t).

The numerical scheme studied in the current paper is a generalization of the scheme presented in [18]. It was applied for the discretization of the conservative two-fluid model with only one momentum equation, therefore the potential φ is a scalar. In addition, the potential φ is independent of the space variable. The main improvement of the current work is to propose a formulation able to deal with space dependent potentials and nonconservative products. Note that the space dependency of the potential is required in the modeling of source terms such as gravity in the isentropic Euler equation or the bathymetry in the shallow water model. The nu- merical strategy is described in a general multidimensional framework R

^d

and does not require a bound for the eigenvalues of the system. The stability of the scheme is ensured under an advective CFL condition, i.e. not restrictive for a continuous solu- tion in the regime of small ε. The main results was already presented in [37] without mathematical proofs.

2. Centered-potential regularization of AUS method. In the current sec-

tion, an adaptation of the so-called Advection Upstream Splitting Method (AUSM)

introduced in [26] is proposed. The numerical strategy, from now on called Centered-

Potential Regularization (CPR) is based on a centered estimate of the potential, as

recommended in [12] to preserve the asymptotic limit, see Section 3. In addition,

the centered estimate ensures the discrete stability of the steady state at rest, see

Section 2.3. Then, the advection operator of the momentum equation is discretized

using an upwind scheme according to the sign of the mass flux.

F ace Cell k f

ρ

ⁿ_i,k

, U

_i,kⁿ

N

^k_f

k

Cell k

_f

ρ

ⁿ_i,k

f

, U

_i,kⁿ

f

Fig. 2.Scheme of the numerical notations for 2D case,d= 2.

For any dimension d ∈ N \ {0}, we consider a tessellation of Ω ⊂ R

^d

, denoted T , composed of star-shaped control volumes. For any control volume k ∈ T , its measure is denoted by |k|, its surface area by |∂k| and the set of its faces by F

k

. In addition, for any face f ∈ F

k

, its surface area is denoted by |f| , the neighbor of k relative to the face f by k

_f

∈ T such that k ∩ k

_f

= f and the unit normal to the face f outward to the control volume k by N

_k^kf

, see Fig. 2. The compactness of the control volume is denoted by `

k

=

_|∂k|^|k|

and the normalized face measure by µ

^k_f

=

_|∂k|^|f|

. The numerical unknowns are approximated at time t

ⁿ⁺¹

= t

ⁿ

+δ

_t

where δ

_t

is the time step at the n

^th

iteration. The following numerical strategy is based on a cell-centered finite volume method, i.e. the numerical unknowns are the approximation of the averaged value of the mass ρ

ⁿ_i,k

> 0 and of the velocity U

_i,kⁿ

∈ R

^d

at the time t

ⁿ

in each control volume k ∈ T . The discretization of the potential is denoted by φ

ⁿ_i,k

:= φ

_i

(ρ

ⁿ_k

, X

_k

) and we use the following notations at the face: 2 (a)

_f

:= a

_k

+ a

_kf

and 2 [a]

^k_k^f

:= a

_kf

− a

_k

. The numerical scheme of the mass conservation (M

_ε^t

) reads

(S

_ε^δt

.a) ρ

ⁿ⁺¹_i,k

= ρ

ⁿ_i,k

− δ

_t

`

k

X

f∈Fk

F

_i,fⁿ⁺¹

· N

_k^kf

µ

^k_f

where the numerical mass flux F

_i,fⁿ⁺¹

is an approximation of

_|f|δ¹

t

R

tⁿ⁺¹ tⁿ

R

f

ρ

i

U

i

dσ dt.

Then, the momentum balance is approximated using an upwind scheme. It reads

(S

_ε^δt

.b)

ρ

ⁿ⁺¹_i,k

U

_i,kⁿ⁺¹

= ρ

ⁿ_i,k

U

_i,kⁿ

− ρ

ⁿ⁺¹_i,k

δ

_t

`

k

X

f∈Fk

φ

ⁿ⁺¹_i

f

ε

²

N

_k^kf

µ

^k_f

− δ

_t

`

k

X

f∈Fk

U

_i,kⁿ

F

_i,fⁿ⁺¹

· N

_k^kf

⁺

− U

_i,kⁿ_f

F

_i,fⁿ⁺¹

· N

_k^kf

−

µ

^k_f

with the positive and the negative part functions defined by 2 (ψ)

^±

= |ψ| ± ψ ≥ 0.

The numerical mass flux is regularized using the centered potential variation, i.e.

(S

_ε^δt

.c) F

_i,fⁿ⁺¹

:= ρ

ⁿ⁺¹_i

U

_iⁿ

f

− γδ

_t

ρ

ⁿ⁺¹_i

`

f

φ

ⁿ⁺¹_i

^kf

k

ε

²

N

_k^kf

.

The regularization parameter γ can be defined locally, at the face, as a function of the flow. The sensitivity of the solution to the numerical regularization parameter is clearly depending on the state variables. In the following, under the following implicit CFL condition

(2.1) δ

t

`

_k

X

f∈Fk

F

_i,fⁿ⁺¹

· N

_k^kf

µ

^k_f

< ρ

ⁿ_i,k

and for γ ≥ 1, an entropic stability result is highlighted, see Theorem 2.3. Note that the numerical scheme (S

_ε^δt

) is a nonlinear IMEX scheme. More precisely, the implicit CFL condition (2.1) can not be directly used to estimate a time step since it depends on the mass (through the potential) at the next time iteration. In addition, the mass conservation (S

_ε^δt

.a) is a nonlinear coupled implicit scheme. One can use a Brouwer argument to show the existence of a fixed point. In practice, the scheme (S

_ε^δt

.a) is solved using a Banach fixed point strategy and the time step is estimated using the previous approximation.

Proposition 2.1 ( Positivity ). Assume that the initial condition ρ

⁰_i,k

are positive.

Then there exists a time step δ

t

such that the CFL condition (2.1) is fulfilled and the mass approximations ρ

ⁿ_i,k

are positive.

Proof. Assume that the mass approximation at time iteration n is positive, i.e.

ρ

ⁿ_i,k

> 0, which is true for the initial condition. The CFL condition (2.1) is clearly satisfied for δ

_t

= 0 and it is not an equality case. Then since the mass approximations ρ

ⁿ⁺¹_i,k

are continuous functions of the time step δ

_t

, there exists a neighborhood of 0 for δ

_t

such that the condition is satisfied. Then by direct estimate, we have

ρ

ⁿ⁺¹_i,k

= ρ

ⁿ_i,k

− δ

t

`

_k

X

f∈Fk

F

_i,fⁿ⁺¹

· N

_k^kf

µ

^k_f

> 2 δ

t

`

_k

X

f∈Fk

F

_i,fⁿ⁺¹

· N

_k^kf

−

µ

^k_f

which leads to the positivity of the mass approximations ρ

ⁿ⁺¹_i,k

.

To compare the implicit CFL condition (2.1) to a classical CFL condition, a more restrictive CFL condition but in the classical form is proposed. The numerical mass flux is bounded by

F

_i,fⁿ⁺¹

· N

_f^k

−

≤

ρ

ⁿ⁺¹_i

U

_iⁿ

f

· N

_k^kf

+ γδ

t

ρ

ⁿ⁺¹_i

`

f

φ

ⁿ⁺¹_i

^kf

k

ε

²

.

Since the sum over the faces is normalized, the CFL condition can be computed by face and leads to a second order polynomial function

V

2δt

`_k

²

+V

1δt

`_k

−1 which is negative when (V

₁

+ V

₂

)

_`^δt

k

< 1. After simplification, we get the following CFL condition

(2.2)

ρ

ⁿ⁺¹_i

U

_iⁿ

f

· N

_k^kf

+ ρ

ⁿ⁺¹_i

f

r

γ 2

[

^φn+1i

]

^kf_k ε²

min

ρ

ⁿ⁺¹_i,k

, ρ

ⁿ⁺¹_i,k

f

δ

_t

min `

k

, `

k_f

< 1 2

which is more restrictive than (2.1). Classical Godunov schemes are generally stable under a CFL condition of the form

U

_iⁿ

· N

_f^k

+

^c_εⁿⁱ

δ_t

`

< Cst see [16]. The celerity of the potential wave (c

_i

)

_1≤i≤L

corresponds to the square root of the eigenvalues of the matrix ρ

_i

H

_ij

. In the limit ε goes to zero, the wave potential is still large c

_i

= O

ε→0

(1) and the classical CFL condition becomes restrictive, i.e.

^δ_`^t

= O

ε→0

(ε).

On the contrary, the main term of the potential is constant in space, i.e. φ

ⁿ_i,k

= φ

ⁿ_i,k

f

+ O

ε→0

ε

²

, see Section 3.2 and the CFL condition of the CPR scheme can be large, i.e.

^δ_`^t

= O

ε→0

(1).

The regularization term in the numerical mass flux plays a role similar to the numerical viscosity introduced in the Lax-Wendroff scheme [25]. In particular, dis- persive oscillations are present in the vicinity of the shock using the CPR scheme (S

_ε^δt

) as well as the Lax-Wendroff scheme. Similarly to the artificial viscosity (denoted by B in [25]), the regularization parameter γ can be increased in order to reduce the amplitude of these oscillations.

2.1. Consistency and accuracy order. In the current section, the consistency of the CPR scheme (S

_ε^δt

) in the sense of finite difference methods is analyzed. The compactness reads `

_k

=

_|∂k|^|k|

=

^δ_2d^x

with d the dimension and δ

_x

is the standard space step definition, i.e. the distance between the center of two neighboring cells. For readability reasons, the space Taylor expansion is written in one direction only. The neighbors of the volume k ∈ Z in the considered direction are k − 1 and k + 1. The notation k +

¹

/

2

indicate a discretization at the face between the volume k and k + 1.

Even if the index of the other directions are not indicated, the proof is established in multi-dimensional framework d ∈ N \ {0}.

Proposition 2.2 ( Consistency ). Assume that the solution is smooth enough and the tessellation is a regular cartesian grid with a space step δ

_x

. Then for any regularization coefficient γ bounded with respect to δ

_t

and δ

_x

the CPR scheme (S

_ε^δt

) is consistent with the advection model (M

_ε^t

). More precisely the modified system reads



 

 

 



 

 

 



∂

t

ρ

i

+ ∇ · (ρ

i

W

i

) = O

δ

_t²

, δ

²_x

, δ

t

δ

_x²

ε

²

∂

t

(ρ

i

U

i

) + ∇ · (ρ

i

U

i

⊗ W

i

) = −ρ

i

∇ φ

i

ε

²

− δ

t

2

∇ · (ρ

i

∂

t

U

i

⊗ U

i

) + ∂

t

ρ

i

ε

²

∇φ

i

+O

δ

²_t

, δ

x

, δ

t

δ

²_x

ε

²

.

with the modified discharge ρ

i

W

i

= ρ

i

U

i

−

^δ₂^t

2dγρ

i

∇

_φ

i

ε²

+ ρ

i

∂

t

U

i

− U

i

∂

t

ρ

i

. Proof. The exact solution of (M

_ε^t

) at time t

ⁿ

and at the position kδ

x

is denoted by ρ

i

|

ⁿ_k

:= ρ

i

(t

ⁿ

, kδ

x

) and U

i

|

ⁿ_k

:= U

i

(t

ⁿ

, kδ

x

). Let us focus firstly on the numerical mass flux (S

_ε^δt

.c). The centered series leads to

1 4

ρ

_i

|

ⁿ⁺¹_k+1

+ ρ

_i

|

ⁿ⁺¹_k

φ

_i

|

ⁿ⁺¹_k+1

− φ

_i

|

ⁿ⁺¹_k

= δ

_x

2 ρ

_i

∂

_x

φ

_i

|

ⁿ⁺¹_k+1/₂

+ δ

_x³

2 C|

ⁿ⁺¹_k+1/₂

+ O δ

⁵_x

= δ

x

2 ρ

i

∂

x

φ

i

|

ⁿ⁺_k+1^1//²₂

+ δ

³_x

2 C|

ⁿ⁺_k+1¹/^/₂²

+ O δ

t

δ

x

, δ

_x⁵

where C =

¹₈

∂

_x²

ρ

i

∂

x

φ

i

+

^∂3x₃^φi

and 1

2

ρ

_i

|

ⁿ⁺¹_k+1

U

_i

|

ⁿ_k+1

+ ρ

_i

|

ⁿ⁺¹_k

U

_i

|

ⁿ_k

= ρ

i

|

ⁿ⁺¹_k+1/₂

U

i

|

ⁿ_k+1/₂

+ δ

²_x

8 ∂

x

ρ

i

|

ⁿ⁺¹_k+1/₂

U

i

|

ⁿ_k+1/₂

+ O δ

_x⁴

= ρ

i

U

i

|

ⁿ⁺_k+1^1//²2

+ δ

t

2 T |

ⁿ⁺_k+1^1//²2

+ δ

²_x

D|

ⁿ⁺_k+1¹/^/2²

+ O δ

²_t

, δ

_x⁴

with D =

¹₈

∂

x

ρ

i

U

i

+

^δ₂^t

T

and T = U

i

∂

t

ρ

i

− ρ

i

∂

t

U

i

. Using a trapezoidal rule, the

flux at time t

ⁿ

+

^δ₂^t

is a second order approximation of the integral of the flux along

the time step. More precisely, we have F

i

|

ⁿ⁺¹_k+1/2

=

ρ

i

W

i

+ δ

_x²

D + δ

t

δ

_x²

ε

²

dγC

n+¹/₂

k+¹/₂

+ O δ

²_t

, δ

_x⁴

with F

i

|

ⁿ⁺¹_k+1/₂

the discrete flux (S

_ε^δt

.c) estimated with the exact solution. Using the centered series, the modified equation of the mass is obtained. Considering the mo- mentum equation, the upwind scheme leads to the following series

U

i

|

ⁿ_k

F

i

|

ⁿ⁺¹_k+1/₂

· N

_k^kf

⁺

− U

i

|

ⁿ_k

f

F

i

|

ⁿ⁺¹_k+1/₂

· N

_k^kf

−

=

U

i

+ δ

_t

2 ∂

t

U

i

ρ

i

W

i

n+¹/2

k+¹/2

+ O δ

²_t

, δ

x

The right-hand-side is finally written to get the modified momentum balance.

2.2. Stability and entropy dissipation. In the current section, a stability result based on the dissipation of the discrete mechanical energy, which stands for a mathematical entropy, is presented. Let us introduce the discrete kinetic energy K

_i,kⁿ

:=

^ρ

n i,k

2

U

_i,kⁿ

2

, the discrete potential energy E

_kⁿ

:= E (ρ

ⁿ_k

, X

k

) and the discrete mechanical energy E

ⁿ_k

:=

^E_ε^kn₂

+ P

L

i=1

K

ⁿ_i,k

.

Theorem 2.3 ( Dissipation of the discrete mechanical energy ). Assume that the CFL condition (2.1) is fulfilled. Then for any regularization coefficient γ ≥ 1, the discrete mechanical energy satisfies the following local entropy inequality

(2.3) E

_kⁿ⁺¹

≤ E

_kⁿ

− δ

t

`

_k

X

f∈Fk

L

X

i=1

G

_K,i,fⁿ⁺¹

+ G

_E,i,fⁿ⁺¹

ε

²

!

· N

_k^kf

+ H

ⁿ⁺¹_K,i,f

+ H

ⁿ⁺¹_E,i,f

ε

²

! µ

^k_f

with the fluxes G

_K,i,fⁿ⁺¹

and H

_K,i,fⁿ⁺¹

of kinetic energy defined by

(2.4)

G

_K,i,fⁿ⁺¹

· N

_k^kf

:= 1 2

U

_i,kⁿ

2

F

_i,fⁿ⁺¹

· N

_k^kf

⁺

− 1 2

U

_i,kⁿ _f

2

F

_i,fⁿ⁺¹

· N

_k^kf

−

and H

ⁿ⁺¹_K,i,f

:= δ

t

ρ

ⁿ⁺¹_i

`

^kf

k

φ

ⁿ⁺¹_i

^kf

k

ε

²

2

and the fluxes G

_E,i,fⁿ⁺¹

and H

ⁿ⁺¹_E,i,f

of potential energy defined by

(2.5)

G

ⁿ⁺¹_E,i,f

· N

_f^k

:= φ

ⁿ⁺¹_i

f

F

_i,fⁿ⁺¹

· N

_k^kf

and H

_Eⁿ⁺¹_,i,f

:= −

φ

ⁿ⁺¹_i

^kf

k

ρ

ⁿ⁺¹_i

U

_iⁿ

^kf

k

· N

_k^kf

.

The proof of the theorem results from two lemmas that establish in the discrete framework the evolution of the kinetic energy and potential energy.

Lemma 2.4 ( Kinetic energy ). Assume that the CFL condition (2.1) is fulfilled . Then the discrete kinetic energy satisfies the following bound

(2.6) K

ⁿ⁺¹_i,k

≤ K

ⁿ_i,k

− δ

t

`

_k

X

f∈Fk

G

_K,i,fⁿ⁺¹

· N

_f^k

+ H

ⁿ⁺¹_K,i,f

− R

ⁿ⁺¹_K,i,f

µ

^k_f

− δ

t

Q

ⁿ⁺¹_i,k

ε

²

where the fluxes of potential energy G

_K,i,fⁿ

and H

ⁿ_K,i,f

are defined by (2.4), the source term and the discrete work of the forces are respectively given by

(2.7) R

ⁿ⁺¹_K,i,f

:= δ

t

ρ

ⁿ⁺¹_i

`

f

φ

ⁿ⁺¹_i

^kf

k

ε

²

2

and Q

ⁿ⁺¹_i,k

:= ρ

ⁿ⁺¹_i,k

U

_i,kⁿ

`

k

· X

f∈Fk

φ

ⁿ⁺¹_i

^kf

k

N

_k^kf

µ

^k_f

.

Proof. According to Proposition 2.1, the masses ρ

ⁿ⁺¹_i,k

are positive. Then the scheme satisfied by the velocity U

_i,kⁿ⁺¹

is obtained by replacing the mass ρ

ⁿ_i,k

in the momentum scheme (S

^δ_ε^t

.b) using the mass scheme (S

^δ_ε^t

.a). More precisely

(2.8)

U

_i,kⁿ⁺¹

= U

_i,kⁿ

+ δ

t

`

_k

X

f∈Fk

U

_i,kⁿ

f

− U

_i,kⁿ

ρ

ⁿ⁺¹_i,k

F

_i,fⁿ⁺¹

· N

_f^k

−

µ

^k_f

− δ

t

`

_k

X

f∈Fk

φ

ⁿ⁺¹_i

f

ε

²

N

_k^kf

µ

^k_f

.

The centered-discretization of the potential at faces can replaced by the half-difference, X

f∈Fk

φ

ⁿ⁺¹_i

f

N

_k^kf

µ

^k_f

= X

f∈Fk

φ

ⁿ⁺¹_i

^kf

k

N

_k^kf

µ

^k_f

.

Using the equality 2A · (B − A) = kBk

²

− kAk

²

− kB − Ak

²

for any A ∈ R

^d

and B ∈ R

^d

, the scalar product between the velocity scheme (2.8) and the ρ

ⁿ⁺¹_i,k

U

_i,kⁿ

leads to

K

ⁿ⁺¹_i,k

= K

ⁿ_i,k

− δ

t

`

_k

X

f∈Fk

G

_K,i,fⁿ⁺¹

· N

_k^kf

µ

^k_f

− δ

t

Q

ⁿ⁺¹_i,k

ε

²

+ S

_i,kⁿ⁺¹

where the source term S

_i,kⁿ⁺¹

coming from the numerical discretization is given by S

_i,kⁿ⁺¹

:= 1

2 ρ

ⁿ⁺¹_i,k

U

_i,kⁿ⁺¹

− U

_i,kⁿ

2

− 2 δ

t

`

_k

X

f∈Fk

[U

_iⁿ

]

^k_k^f

2

F

_i,fⁿ⁺¹

· N

_f^k

−

µ

^k_f

.

The following step of the proof looks for an upper bound the source term S

_i,kⁿ⁺¹

in function of the potential variation. Using the velocity scheme (2.8) and Jensen’s inequality, the advection term and the variation of potential are split, i.e.

S

ⁿ⁺¹_i,k

≤ ρ

ⁿ⁺¹_i,k

δ

t

`

k

2

X

f∈Fk

φ

ⁿ⁺¹_i

^kf

k

ε

²

2

µ

^k_f

+ 1 ρ

ⁿ⁺¹_i,k

δ

t

`

k

²

X

f∈Fk

r

F

_i,fⁿ⁺¹

· N

_f^k

−

! 2 [U

_iⁿ

]

^k_k^f

r

F

_i,fⁿ⁺¹

· N

_f^k

−

! µ

^k_f

2

−2 δ

_t

`

k

X

f∈Fk

[U

_iⁿ

]

^k_k^f

2

F

_i,fⁿ⁺¹

· N

_f^k

⁻

µ

^k_f

.

Then the first term of the right-hand-side is split into an anti-symmetric part which

leads to the flux H

ⁿ⁺¹_K,i,f

given by (2.4) and the symmetric part which leads to a source

term R

ⁿ⁺¹_K,i,f

estimated at the faces (2.7). Finally, using a Cauchy-Schwarz’s inequality,

the last terms of the right-hand-side is bounded by S

_i,kⁿ⁺¹

≤ δ

t

`

k

X

f∈Fk

R

ⁿ⁺¹_K,i,f

− H

ⁿ⁺¹_K,i,f

µ

^k_f

+4 δ

t

`

k



 X

f∈Fk

[U

_iⁿ

]

^k_k^f

2

F

_i,fⁿ⁺¹

· N

_k^kf

−

µ

^k_f









 δ

t

`

k

X

f∈Fk

F

_i,fⁿ⁺¹

· N

_k^kf

−

ρ

ⁿ⁺¹_i,k

µ

^k_f

− 1 2





 .

We conclude using the CFL condition (2.1).

Lemma 2.5 ( Potential energy ). The discrete potential energy satisfies the follow- ing bound

(2.9) E

_kⁿ⁺¹

≤ E

_kⁿ

− δ

_t

`

k

X

f∈Fk

L

X

i=1

G

_Eⁿ⁺¹_,i,f

· N

_f^k

+ H

_Eⁿ⁺¹_,i,f

− R

_Eⁿ⁺¹_,i,f

µ

^k_f

+ δ

_t

L

X

i=1

Q

ⁿ⁺¹_i,k

where the fluxes of potential energy G

_E,i,fⁿ

and H

_Eⁿ_,i,f

are defined by (2.5) and the work of the force Q

ⁿ_i,k

is given by (2.7). The numerical source term R

ⁿ⁺¹_E,i,f

is defined by (2.10) R

_Eⁿ⁺¹_,i,f

:=

φ

ⁿ⁺¹_i

^kf

k

F

_i,fⁿ⁺¹

− ρ

ⁿ⁺¹_i

U

_iⁿ

f

· N

_k^kf

.

Proof. Let us consider the potential energy at the intermediate time step E

_k^n+s

= E sρ

ⁿ⁺¹_k

+ (1 − s) ρ

ⁿ_k

, X

k

. Accordingly to the mean value theorem, there exists 0 <

s

^?_k

< 1 such that

E

_kⁿ⁺¹

= E

_kⁿ

+ ∂

s

E

_kⁿ⁺¹

− 1

2 ∂

_ss²

E

_k^n+s?k

= E

_kⁿ

+

L

X

i=1

ρ

ⁿ⁺¹_i,k

− ρ

ⁿ_i,k

φ

ⁿ⁺¹_i,k

− 1 2

L

X

i=1 L

X

j=1

ρ

ⁿ⁺¹_i,k

− ρ

ⁿ_i,k

H

^n+s

? k

ij,k

ρ

ⁿ⁺¹_j,k

− ρ

ⁿ_j,k

with H

^n+s_ij,k^?k

= H

_ij

s

^?_k

ρ

ⁿ⁺¹_k

+ (1 − s

^?_k

) ρ

ⁿ_k

, X

_k

. Under the assumption of (1.1), the last term is nonnegative. Then, using the mass scheme (S

_ε^δt

.a), the following bound holds

E

_kⁿ⁺¹

≤ E

_kⁿ

− δ

_t

`

k

X

f∈Fk

L

X

i=1

φ

ⁿ⁺¹_i,k

F

_i,fⁿ⁺¹

· N

_k^kf

µ

^k_f

.

Finally, the exchange of energy Q

ⁿ⁺¹_i,k

is added to the right-hand-side and the rest is split into an anti-symmetric flux G

_E,i,fⁿ⁺¹

and a symmetric residual R

ⁿ⁺¹_E,i,f

to get the announced result.

Proof. [of Theorem 2.3] The definition of the numerical mass flux (S

_ε^δt

.c) is mo- tivated by the leading form of the potential energy residual (2.10). More precisely, injecting (S

_ε^δt

.c) in (2.10) and summing with (2.7), the mechanic source term reads

R

ⁿ⁺¹_K,i,f

+ R

ⁿ⁺¹_E,i,f

ε

²

= (1 − γ) δ

t

ρ

ⁿ⁺¹_i

`

f

φ

ⁿ⁺¹_i

^kf

k

ε

²

2

.

We conclude the dissipation law of the mechanical energy (2.3).

Note that the bound (2.3) is a dissipation law since the fluxes G

_K,i,fⁿ⁺¹

· N

_k^kf

, G

_E,i,fⁿ⁺¹

· N

_k^kf

, H

ⁿ⁺¹_K,i,f

and H

ⁿ⁺¹_E,i,f

are anti-symmetric with respect to the control volume. More precisely, summing inequality (2.3) over the cells k ∈ T leads to the energy decay law, which can be considered as a nonlinear stability argument. The mechanical energy dissipation leads to the following bound of the variation of potential, i.e.

(γ − 1)

T

X

n=1

δ

t

X

f∈F

ρ

ⁿ_i

`

f

[φ

ⁿ_i

]

^k_k^f

ε

²

2

≤ X

k∈T

`

k

E

_k⁰

.

2.3. Conservation law and steady states at rest. The following section is devoted to the translation of the physical properties Proposition 1.1 and Proposi- tion 1.2 of the model (M

_ε^t

) at the discrete level.

Proposition 2.6 ( Discrete conservation of the mass ). The discrete mass is conserved by the CPR scheme (S

_ε^δt

). More precisely

X

k∈T

|k|ρ

ⁿ_i,k

= X

k∈T

|k|ρ

⁰_i,k

.

The total momentum conservation Proposition (1.2) is not easy to satisfy at the discrete level for any potential mapping. The following result Proposition (2.7) shows the conservation in the case of a quadratic energy and as a consequence with constant Hessian. However, many physical models have a quadratic energy. For example, the multilayer shallow water model satisfies this assumption and more precisely the Hes- sian is given by H

ij

=

_% ^g

max(i,j)

.

Proposition 2.7 ( Discrete conservation of the total momentum ). Assume that the Hessian is a constant and symmetric matrix, i.e.

∂

_ρi

H = 0, ∇H = 0 and H

_ij

= H

_ji

. Then the discrete total momentum satisfies the following balance

L

X

i=1

ρ

ⁿ⁺¹_i,k

U

_i,kⁿ⁺¹

=

L

X

i=1

ρ

ⁿ_i,k

U

_i,kⁿ

− δ

t

`

_k

X

f∈Fk

L

X

i=1

ρ

ⁿ⁺¹_i

f

φ

ⁿ⁺¹_i

f

− 1

2 ρ

ⁿ⁺¹_i

φ

ⁿ⁺¹_i

f

µ

^k_f

− δ

t

`

_k

X

f∈Fk

L

X

i=1

U

_i,kⁿ

F

_i,fⁿ⁺¹

· N

_f^k

+

− U

_i,kⁿ_f

F

_i,fⁿ⁺¹

· N

_f^k

−

µ

^k_f

.

It follows that the discrete global momentum is conserved by the CPR scheme (S

_ε^δt

).

More precisely

X

k∈T L

X

i=1

|k|ρ

ⁿ_i,k

U

_i,kⁿ

= X

k∈T L

X

i=1

|k|ρ

⁰_i,k

U

_i,k⁰

.

Proof. Summing the momentum scheme (S

_ε^δt

.b) over the mass 1 ≤ i ≤ L, the nonconservative term becomes

L

X

i=1

ρ

ⁿ⁺¹_i,k

X

f∈Fk

φ

ⁿ⁺¹_i

f

ε

²

N

_f^k

µ

^k_f

= X

f∈Fk

L

X

i=1

ρ

ⁿ⁺¹_i,k

φ

ⁿ⁺¹_i,k

f

2ε

²

N

_f^k

µ

^k_f

Since the Hessian is constant, the potential can be written under the form φ

ⁿ⁺¹_i,k

f

=

P

L

j=1

H

ij

ρ

ⁿ⁺¹_j,k

f

and since the Hessian is symmetric, i.e. H

ij

= H

ji

, it reads

L

X

i=1

ρ

ⁿ⁺¹_i,k

φ

ⁿ⁺¹_j,k

f

=

L

X

i=1 L

X

j=1

ρ

ⁿ⁺¹_i,k

H

_ij

ρ

ⁿ⁺¹_j,k

f

=

L

X

i=1 L

X

j=1

ρ

ⁿ⁺¹_j,k

f

H

_ji

ρ

ⁿ⁺¹_i,k

=

L

X

j=1

ρ

ⁿ⁺¹_j,k

f

φ

ⁿ⁺¹_j,k

.

Then the half sum of the left-hand side and the right-hand side leads to

L

X

i=1

ρ

ⁿ⁺¹_i,k

X

f∈Fk

φ

ⁿ⁺¹_i

f

ε

²

N

_f^k

µ

^k_f

= X

f∈Fk

L

X

i=1

ρ

ⁿ⁺¹_i,k

φ

ⁿ⁺¹_i,k

f

+ ρ

ⁿ⁺¹_i,k

f

φ

ⁿ⁺¹_i,k

4ε

²

N

_f^k

µ

^k_f

.

Finally using the equality

^ρ

n+1 i,k φⁿ⁺¹

i,kf+ρⁿ⁺¹

i,kfφⁿ⁺¹_i,k

4

= ρ

ⁿ⁺¹_i

f

φ

ⁿ⁺¹_i

f

−

¹₂

ρ

ⁿ⁺¹_i

φ

ⁿ⁺¹_i

f

, we get the announced result.

Proposition 2.8 ( Well-balanced ). The CPR scheme (S

_ε^δt

) preserves the discrete steady states at rest defined by U

_i,kⁿ

= 0 and φ

ⁿ_i,k

= φ

_i

.

Proof. Assume that the discrete unknowns at the n

^th

time iteration satisfy the the steady state at rest. Let us consider first the mass scheme (S

_ε^δt

.a). Since the discharge and the variation of potential vanish ρ

ⁿ⁺¹_i

U

_iⁿ

f

= 0 and [φ

ⁿ_i

]

^k_k^f

= 0, the scheme can be written as

ρ

ⁿ⁺¹_i,k

− γ δ

_t²

`

_k

X

f∈Fk

ρ

ⁿ⁺¹_i

`

f

φ

ⁿ⁺¹_i

^kf

k

ε

²

µ

^k_f

= ρ

ⁿ_i,k

− γ δ

_t²

`

_k

X

f∈Fk

ρ

ⁿ_i

`

f

[φ

ⁿ_i

]

^k_k^f

ε

²

µ

^k_f

.

The unique solution is ρ

ⁿ⁺¹_i,k

= ρ

ⁿ_i,k

, which implies φ

ⁿ⁺¹_i,k

= φ

_i

. Finally, the velocity at time iteration n + 1 vanishes U

_i,kⁿ⁺¹

= 0 since the numerical flux F

_i,fⁿ⁺¹

and the potential variation

φ

ⁿ⁺¹_i

^kf

k

vanish.

Let us highlight that the CPR scheme (S

_ε^δt

) is stable and consistent without condition on the velocity difference. Indeed, for several physical model under the form (M

_ε^t

), an additional condition on the velocity difference is required to proof the hyperbolicity of the system, see [13, 30, 31] in the framework of free surface flows.

Obviously if this condition is not satisfied, the solution of the CPR scheme (S

_ε^δt

) is well-defined but is not relevant as a mathematical solution of the PDE system.

3. Asymptotic regimes. In the following section, the behavior of the scheme in the regime ε 1 is analyzed. Fig. 1 illustrate the following properties. The advection model (M

_ε^t

) has two asymptotic behaviors relevant depending on the time scale considered.

3.1. Fine time scale: the acoustic regime (M

₀^τ

). In the following section, the behavior of the solution of (M

_ε^t

) for a small time scale t = ετ and in the limit where ε goes to zero is analyzed. We refer to [43] for the derivation and the analytical results about the acoustic regime.

For any fixed φ ∈ R

^L

, the main masses ρ

_k

are defined such that φ

_i

(ρ

_k

, X

_k

) = φ

_i

and we consider the following scheme (S

₀^δτ

)

φ e

ⁿ⁺¹_i,k

= φ e

ⁿ_i,k

− δ

τ

`

k L

X

j=1

H

ij

k

X

f∈Fk

ρ

_j

U e

_jⁿ

f

· N

_k^kf

− γδ

τ

ρ

_j

`

f

h φ e

ⁿ⁺¹_j

i

^kf

k

! µ

^k_f

U e

_i,kⁿ⁺¹

= U e

_i,kⁿ

− δ

τ

`

k

X

f∈Fk

φ e

ⁿ⁺¹_i

f

N

_k^kf

µ

^k_f

with H

_ij

k

:= H

_ij

(ρ

_k

, X

_k

). The following consistency result holds

Proposition 3.1. Assume that the solution is smooth enough and the tessellation is a regular cartesian grid with a space step δ

x

. Then the numerical strategy (S

₀^δτ

) is consistent with the acoustic regime (M

₀^τ

). More precisely the modified system reads



 

 



 

 



∂

τ

φ e

i

+

L

X

j=1

H

ij

∇ · ρ

_j

U e

j

= δ

τ

2

L

X

j=1

H

ij

∇ ·

(2dγ − 1) ρ

_j

∇ φ e

j

+ O δ

_τ²

, δ

²_x

∂

_τ

U e

_i

+ ∇ φ e

_i

= − δ

τ

2

L

X

j=1

∇

H

_ij

∇ · ρ

_j

U e

_j

+ O δ

²_τ

, δ

_x²

.

Proof. The proof is similar to the proof of Proposition 2.2.

Proposition 3.2. Assume that the Hessian of the potential energy is well- conditioned independently of ε, i.e. κ = O

ε→0

(1), and the discretized initial condition satisfies

ρ

⁰_i,k

= O

ε→0

(1) , U

_i,k⁰

= O

ε→0

(ε) and φ

i

ρ

⁰_k

, X

k

= φ

_i

+ O

ε→0

(ε) . with the parameter ρ

_i,k

such that the potential φ

_i

= φ

_i

(ρ

_k

, X

_k

) is space independent.

Then the CPR scheme (S

^δ_ε^t

) with the time step scaling δ

_t

= O

ε→0

(ε) tends to the scheme (S

₀^δτ

) in the limit where ε goes to zero. More precisely, for any time iteration n ∈ N , for any control volume k ∈ T scheme differences read

ρ

ⁿ_i,k

− ρ

_i,k

+ ε ρ e

ⁿ_i,k

= O

ε→0

ε

²

and U

_i,kⁿ

− ε U e

_i,kⁿ

= O

ε→0

ε

²

with

ρ

ⁿ_i,k

, U

_i,kⁿ

, the solution of the CPR scheme (S

_ε^δt

) and ρ e

ⁿ_k

= H

⁻¹_k

φ e

ⁿ_k

with

φ e

ⁿ_i,k

, U e

_i,kⁿ

, the solution of the scheme (S

₀^δτ

) with the initial condition ρ e

⁰_i,k

= ρ

⁰_i,k

, U e

_i,k⁰

= U

_i,k⁰

and H

⁻¹_k

is the inverse matrix of H

k

i,j

= ∂

ρ_j

φ

i

(ρ

_k

, X

k

).

Proof. First the time of the numerical scheme is rescaled. Note that since U

_i,kⁿ

is a velocity, the rescaled velocity becomes u

ⁿ_i,k

:=

^U

n i,k

ε

. Assume the following scaling of the numerical unknowns at the previous iteration n

φ

ⁿ_i,k

= φ

_i

(ρ

ⁿ_k

, X

_k

) = φ

_i

+ O

ε→0

(ε) and u

ⁿ_i,k

= O

ε→0

(1)

which is true by hypothesis for the initial condition. The main term of the mass scheme (S

_ε^δt

.a) (with δ

τ

:=

^δ_ε^t

) can be written as

ρ

ⁿ⁺¹_i,k

− γ δ

_τ²

`

_k

X

f∈Fk

ρ

ⁿ⁺¹_i

`

f

φ

ⁿ⁺¹_i

^kf

k

µ

^k_f

= ρ

ⁿ_i,k

− γ δ

_τ²

`

_k

X

f∈Fk

ρ

ⁿ_i

`

f

[φ

ⁿ_i

]

^k_k^f

µ

^k_f

+ O

ε→0

(ε) .